Pythagoras' Theorem

Learning Objective:

• we are learning to find a missing side length in a right-angled triangle using Pythagoras’ Theorem.

Investing:

• This is useful because it will allow us to find the length of a side in a right-angled triangle
• A functional (real-life) application is construction: building things.

• There's really good evidence that people have known about this relationship between the side lengths in right angled triangles for four and a half thousand years

• A 'free gift' with this skill is that we can now start to label a right-angled triangle for trigonmetry
• This could help you if you want to work in construction

Preparing:

• Are we ready? Can you already identify the hypotenuse on a right angled triangle?
• Let's be sure you can identify the hypotenuse and then label the sides of a right angled triangle
• Before you start you need to know that the hypotenuse of a right angled triangle is the longest side and that this can always be found opposite the right angle
• Key words: hypotenuse, Pythagoras, right-angled triangle
  
  
• You will have a deeper understanding if you also know how to rearrange formulae and square root to undo squaring

Discovering:

• Can you figure it out yourself from these examples?

• What do you notice about the areas of the squares?
• If you set the width to 8 and height to 6, what will the three square areas be?
• So what length will the hypotenuse c be in this triangle?

• The sliders won't allow it, but suppose I could set a width a = 12 and height b = 5.
• What would the hypotenuse c be now? Why?

• What if I set a = 7 and b = 24?

• Can you write a general rule about how c is connected to a and b?
• Can you write that rule in algebra?

• In this Geogebra worksheet you can explore a little further

Modeling:

• add a video example here?
• show some screenshots of correct solutions to simple Pythagoras' problems

• Show a diagram, get pupils to use MWB showing which side is the hypotenuse and all their working out without the answer.

• Here are some examples of people getting it wrong in typical ways:
  • Look for misconceptions when finding one of the shorter sides.
  • Show two triangles the with the same dimensions and the same calculations, one finding the hypotenuse with the wrong calculation, one finding one of the shorter sides with the correct calculation

• Here are some more examples. Did they get it right or wrong? Explain how you know!

Discussing:

• What would this one be? Tell your learning partner. Convince them you're right.
• Explain how you know.
• How would you explain this to someone who was new to it?

Explaining:

• One way to do this is...
• Another approach might be...
• A useful shortcut is to...
• This works because...
• It doesn't work when...
• An exception is...
• Watch out for...
• A common mistake is...
• You can check your result by...
• We can prove this works by...

Practicing:

• Some straightforward examples.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.
• You can demonstrate fluency by at least...

Sharing:

• Now you've learnt about Pythagoras' theorem make a web page or wiki ted to explain it to another learner
• Now you've learnt about Pythagoras' theorem make a (PowerPoint or Keynote) presentation and post it online
• Now you've learnt about Pythagoras' theorem make a poster or model that shows how it works

Assessing:

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...

Developing:

• Next we could learn...
• This leads to...
• Now try...